Simple and Explicit Estimating Functions for a Discretely Observed Diffusion Process

We consider a one-dimensional diffusion process X , with ergodic property, with drift b ( x , θ) and diffusion coefficient a ( x , θ) depending on an unknown parameter θ that may be multidimensional. We are interested in the estimation of θ and dispose, for that purpose, of a discretized trajectory, observed at n equidistant times t_i = iΔ , i = 0, ..., n . We study a particular class of estimating functions of the form ∑ f (θ, X _{t i −1}) which, under the assumption that the integral of f with respect to the invariant measure is null, provide us with a consistent and asymptotically normal estimator. We determine the choice of f that yields the estimator with minimum asymptotic variance within the class and indicate how to construct explicit estimating functions based on the generator of the diffusion. Finally the theoretical study is completed with simulations.     

On some dependence structures for multidimensional Lévy driven moving averages

The Lévy copula can describe the dependence structure of a multidimensional Lévy process or a multivariate infinitely divisible random variable. Suppose the Lévy copula of a multidimensional Lévy process is known. We present the Lévy copula of the Lévy measure of the moving average driven by the multidimensional Lévy process. If there exist some special dependence structures among the components of the Lévy process, we give some dependence invariance properties after the transform of the moving average.     

New invariant and consistent chi-squared type goodness-of-fit tests for multivariate normality and a related comparative simulation study

ABSTRACT

New invariant and consistent goodness-of-fit tests for multivariate normality are introduced. Tests are based on the Karhunen–Loève transformation of a multidimensional sample from a population. A comparison of simulated powers of tests and other well-known tests with respect to some alternatives is given. The simulation study demonstrates that power of the proposed McCull test almost does not depend on the number of grouping cells. The test shows an advantage over other chi-squared type tests. However, averaged over all of the simulated conditions examined in this article, the Anderson–Darling type and the Cramer–von Mises type tests seem to be the best.     

Stereological formulas for stationary surface processes

In the paper surface processes are considered, i.e. sets of surfaces distributed at random in the space. For translation invariant surface processes the PALM distribution of the direction of the normal in a typical surface point is defined and related to two roses of intersection. Fibre and point processes are studied which result by intersections with planes and lines respectively. For these processes stereological formulas are proved which connect some quantities with such of the surface process. As a special ease motion invariant, i.e. translation invariant and isotropic, surface processes are treated, Furthermore, weighted surface processes are considered.     

Generalized discriminant analysis based on distances

This paper describes a method of generalized discriminant analysis based on a dissimilarity matrix to test for differences in a priori groups of multivariate observations. Use of classical multidimensional scaling produces a low‐dimensional representation of the data for which Euclidean distances approximate the original dissimilarities. The resulting scores are then analysed using discriminant analysis, giving tests based on the canonical correlations. The asymptotic distributions of these statistics under permutations of the observations are shown to be invariant to changes in the distributions of the original variables, unlike the distributions of the multi‐response permutation test statistics which have been considered by other workers for testing differences among groups. This canonical method is applied to multivariate fish assemblage data, with Monte Carlo simulations to make power comparisons and to compare theoretical results and empirical distributions. The paper proposes classification based on distances. Error rates are estimated using cross‐validation.     

Parameter estimation in semi-linear models using a maximal invariant likelihood function

In this paper, we consider the problem of estimation of semi-linear regression models. Using invariance arguments, Bhowmik and King [2007. Maximal invariant likelihood based testing of semi-linear models. Statist. Papers 48, 357–383] derived the probability density function of the maximal invariant statistic for the non-linear component of these models. Using this density function as a likelihood function allows us to estimate these models in a two-step process. First the non-linear component parameters are estimated by maximising the maximal invariant likelihood function. Then the non-linear component, with the parameter values replaced by estimates, is treated as a regressor and ordinary least squares is used to estimate the remaining parameters. We report the results of a simulation study conducted to compare the accuracy of this approach with full maximum likelihood and maximum profile-marginal likelihood estimation. We find maximising the maximal invariant likelihood function typically results in less biased and lower variance estimates than those from full maximum likelihood.     

Extension to higher dimensions of the jaeschke-eicker result on the standardized empirical process

The asymptotic distribution of the sup-norm of the heavily weighted empirical process is established in the multidimensional case. This theorem extends in particular the famous result in Jaeschke (1975, 1979) to higher dimensions. There is a striking difference between the behaviour for higher dimensions and that for dimension one, especially the limiting distribution is now a simple transformation of a standard exponential random variable.     

Testing and Confidence estimation of the Mean of a multidimensional Gaussian Process

From the sequential observation of a multidimensional continuous time Gaussian process, whose mean vector depends linearly of a multidimensional parameter, we consider the confidential estimation of the parameter value and the testing problem of a simple hypothesis about the parameter, in presence of a nuisance variance parameter. The method is based on a previously obtained [cf. 4] point estimate for the case of a known covariance structure. We first see that this estimate is, in fact, independent of the variance parameter. For the hypotheses testing problem, the invariance under certain groups of transformations and the partial sufficiency allows to construct optimal terminal tests. Furthermore we determine the observation time necessary to control its power function. These testing results may be translated in terms of most accurate confidence sets. If the observation is stopped according to the diameter of the confidence set, under some condition, the confidence level is preserved.     

BERNOULLI SAMPLING

The concepts of the Bernoulli count process of a point process and Bernoulli sampling of a discrete parameter stochastic process are introduced. The Bernoulli count process determines the stochastic structure of the point process, and a process obtained by thinning a discrete parameter stochastic process by Bernoulli sampling satisfies the same property. Stationarity and the Markov property remain invariant under Bernoulli sampling.     

LAN property for an ergodic diffusion with jumps

In this paper, we consider a multidimensional ergodic diffusion with jumps driven by a Brownian motion and a Poisson random measure associated with a compound Poisson process, whose drift coefficient depends on an unknown parameter. Considering the process discretely observed at high frequency, we derive the local asymptotic normality (LAN) property.     

Relative Price Changes and Inequality in the Size Distribution of Various Components of Income

This article uses a comprehensive model of economic inequality to examine the impact of relative price changes on inequality in the marginal distributions of various income components in which the marginal distributions are derived from a multidimensional joint distribution. The multidimensional joint distribution function is assumed to be a member of the Pearson Type VI family; that is, it is assumed to be a beta distribution of the second kind. The multidimensional joint distribution is so called because it is a joint distribution of components of income and expenditures on various commodity groups. Gini measures of inequality are devised from the marginal distributions of the various income components. The inequality measures are shown to depend on the parameters of the multidimensional joint distribution. It is then shown that the parameters of the multidimensional joint distribution depend on the relative prices of various commodity groups and several other specified exogenous variables. Thus, knowledge of how changes in relative prices affect the parameters of the multidimensional joint distribution is deductively equivalent to knowledge of how changes in relative prices affect inequality in the marginal distributions of various components of income. It is found that relative price changes have a statistically significant impact on inequality in various components of income.     

Bayes estimates of estimable parameters with a direchlet invariant process

The Bayes estimates of estimable parameters of degree 2 are obtained against a Dirichlet invariant process prior and the squared error loss. Some examples are given with the limits of the Bayes estimates.     

Multidimensional scaling used in multivariate statistical process control

This paper considers the use of multidimensional scaling techniques in multivariate statistical process control. Principal components analysis, multiple principal components analysis, partial least squares and PARAFAC models have already been established as useful methods for such, but it should be possible to widen the portfolio of techniques to include others that come under the multidimensional scaling class. Some of these are briefly described-namely classical scaling, non-metric scaling, biplots, Procrustes analysis-and are then used on some gas transportation data provided by Transco.     

Multidimensional scaling used in multivariate statistical process control

This paper considers the use of multidimensional scaling techniques in multivariate statistical process control. Principal components analysis, multiple principal components analysis, partial least squares and PARAFAC models have already been established as useful methods for such, but it should be possible to widen the portfolio of techniques to include others that come under the multidimensional scaling class. Some of these are briefly described-namely classical scaling, non-metric scaling, biplots, Procrustes analysis-and are then used on some gas transportation data provided by Transco.     

Multidimensional Scaling for Selecting Small Groups in College Courses

Many college courses use group work as a part of the learning and evaluation process. Class groups are often selected randomly or by allowing students to organize groups themselves. However, if it is desired to control some aspect of the group structure, such as increasing schedule compatibility within groups, multidimensional scaling can be used to form such groups. This article describes how this has been adopted in an undergraduate statistics course. Resulting groups have been more homogeneous with respect to student schedules than groups selected randomly—an example from winter quarter 2004 increased correlations between student schedules from a mean of .29 before grouping to a within-group mean of .50. Further, the exercise allows opportunities to discuss a wealth of statistical concepts in class, including surveys, association measures, multidimensional scaling, and statistical graphics.     

Transient detailed balance and product form for reaction networks

This paper explores the boundary of the set of reaction networks that have an exact transient (truncated) multidimensional Poisson or product-form distribution for the number of particles of different types. Motivated by the birth–death process, we introduce the notions of transient detailed balance and delay functions, and use these notions to obtain the novel transient product-form distribution in a coagulation-fragmentation process for polymers with a tree-like structure from that of the pure coagulation process.     

Some results for multidimensional stationary independent increment processes

The characteristic function, cumulants and moments of vector-valued multidimensional processes, satisfying properties similar to stationary independent increments, are derived. By considering a set of additional postulates for such processes, it is shown that the marginal distribution of such processes is multivariate Poisson. Some of the results in this paper are extensions of the properties of the first two moments of a univariate one-dimensional process with stationary independent increments.     

多维极化的测度:新的方法及其应用

鉴于现有多维极化测度方法未能体现维度之间的相关性,本文引入余弦相似度,并基于相对剥夺理论提出了新的多维极化测度方法。该方法还可以进行静态和动态分解,能够分析每个维度及其变动对多维极化的贡献率。运用中国家庭追踪调查(CFPS)数据,本文从收入、财产、交通通信支出和受教育水平四个维度实证分析了我国城乡居民家庭之间的多维极化。实证结果表明,城乡长期多维极化呈现明显下降趋势,其中教育、交通通信支出和收入三个维度的极化效应是主要原因,而财产极化效应不利于多维极化的下降。     

Characterization of discrete scale invariant Markov sequences

ABSTRACT

Some special sampling of discrete scale invariant (DSI) processes are presented to provide a multi-dimensional self-similar process in correspondence. By imposing Markov property we show that the covariance functions of such Markov DSI sequences are characterized by variance, and covariance of adjacent samples in the first scale interval. We also provide a theoretical method for estimating spectral density matrix of corresponding multi-dimensional self-similar Markov process. Some examples such as simple Brownian motion (sBm) with drift and scale invariant autoregressive model are presented and these properties are investigated. We present two new method to estimate Hurst parameter of DSI processes and apply them to some sBm and also to the SP500 indices for some period which has DSI property. We compare our estimates with the maximum-likelihood and rescaled range (R/S) method which are applied to the corresponding multi-dimensional self-similar processes.     

A note on existence and construction of invariant loss functions

In a statistical decision problem, if the model is invariant under a transformation group, it is desirable or even compelling to apply equivariance for choosing a decision rule. However, formal equivariance also requires an invariant loss function. In this paper, we give a necessary and sufficient condition for the existence of invariant loss functions, and characterize all invariant loss functions, when the condition is satisfied. Analogous results for the more general case, where the quantity of inferential interest depends also on the observed data, are presented. We also discuss connections among our results and the equivariance literature and present some illustrative examples.